we define a Hermitian metric \(H\)  on \(\det E\)  by

H(u,v):=P(E*z) (z(u),z(v) ) zr-1,  for \(u \textup { and } v\in \det E_z\) .

By Berndtsson's theorem [1]}, this metric \(H\)  is Griffiths positive, but it is the inequality that leads to this fact we will use. We follow the presentation in [2]} (see also [3]}). Denote the curvature of \(H\)  by \(\Theta \) . Fix \(z\in X\) , \(v\in \det E_z\) , and \(\eta \in T^{1,0}_z X\) . For a local holomorphic frame of \(E^*\)  around \(z\) , we denote by \(\lbrace \zeta _i\rbrace \)  the fiber coordinates with respect to this frame, and by \(\lbrace z_\alpha \rbrace \)  the local coordinates on \(X\) . Around \(P(E^*_z)\)  in \(P(E^*)\) , we have homogeneous coordinates \([\zeta _1,...,\zeta _r]\)  which induce local coordinates \((w_1,...,w_{r-1})\) . For a local frame \(e^*\)  of \(O_{P(E^*)}(1)\) , we denote \(h(e^*,e^*)\)  by \(e^{-\phi }\)  and write the tangent vector \(\eta =\sum \eta _\alpha \partial /\partial z_\alpha \) . The inequality that leads to Berndtsson's theorem is
\(-H(\Theta v,v)(\eta ,\bar{\eta })\le \int _{P(E^*_z)} \rho (\Psi _{z}(v),\Psi _{z}(v) ) r \sum _{\alpha ,\beta } \big ( \sum _{i,j}\phi _{\alpha \bar{j}}\phi ^{i\bar{j}}\phi _{i\bar{\beta }}-\phi _{\alpha \bar{\beta }} \big )\eta _\alpha \bar{\eta }_\beta \omega _z^{r-1},\) 
